LMFDB - Embedded newform 6845.2.a.k.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6845,2,Mod(1,6845)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6845.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6845, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6845 = 5 \cdot 37^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6845.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,0,2,8,-7,2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$54.6576001836$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 11x^{5} - x^{4} + 35x^{3} + 7x^{2} - 27x - 9 $ x^7 - 11x^5 - x^4 + 35x^3 + 7x^2 - 27x - 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 185)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.12084$ of defining polynomial
Character	$\chi$	$=$	6845.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.12084 q^{2} +3.33999 q^{3} -0.743718 q^{4} -1.00000 q^{5} -3.74360 q^{6} +1.22417 q^{7} +3.07527 q^{8} +8.15556 q^{9} +1.12084 q^{10} -3.59983 q^{11} -2.48401 q^{12} -0.699609 q^{13} -1.37210 q^{14} -3.33999 q^{15} -1.95945 q^{16} -0.578769 q^{17} -9.14107 q^{18} -7.75609 q^{19} +0.743718 q^{20} +4.08873 q^{21} +4.03484 q^{22} +4.47887 q^{23} +10.2714 q^{24} +1.00000 q^{25} +0.784150 q^{26} +17.2195 q^{27} -0.910439 q^{28} -3.04867 q^{29} +3.74360 q^{30} -7.98457 q^{31} -3.95431 q^{32} -12.0234 q^{33} +0.648708 q^{34} -1.22417 q^{35} -6.06543 q^{36} +8.69333 q^{38} -2.33669 q^{39} -3.07527 q^{40} +5.58669 q^{41} -4.58281 q^{42} -4.02235 q^{43} +2.67726 q^{44} -8.15556 q^{45} -5.02010 q^{46} -8.09571 q^{47} -6.54454 q^{48} -5.50140 q^{49} -1.12084 q^{50} -1.93309 q^{51} +0.520312 q^{52} -9.93959 q^{53} -19.3003 q^{54} +3.59983 q^{55} +3.76466 q^{56} -25.9053 q^{57} +3.41707 q^{58} -0.485356 q^{59} +2.48401 q^{60} +0.919473 q^{61} +8.94942 q^{62} +9.98382 q^{63} +8.35104 q^{64} +0.699609 q^{65} +13.4763 q^{66} +11.2569 q^{67} +0.430441 q^{68} +14.9594 q^{69} +1.37210 q^{70} -0.507354 q^{71} +25.0805 q^{72} -3.55519 q^{73} +3.33999 q^{75} +5.76834 q^{76} -4.40682 q^{77} +2.61906 q^{78} +3.81414 q^{79} +1.95945 q^{80} +33.0464 q^{81} -6.26179 q^{82} -11.6109 q^{83} -3.04086 q^{84} +0.578769 q^{85} +4.50841 q^{86} -10.1825 q^{87} -11.0705 q^{88} -14.3837 q^{89} +9.14107 q^{90} -0.856443 q^{91} -3.33102 q^{92} -26.6684 q^{93} +9.07400 q^{94} +7.75609 q^{95} -13.2074 q^{96} +10.2579 q^{97} +6.16619 q^{98} -29.3586 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{3} + 8 q^{4} - 7 q^{5} + 2 q^{6} - 3 q^{8} + 13 q^{9} - q^{11} - 6 q^{12} + 4 q^{13} - 10 q^{14} - 2 q^{15} - 2 q^{16} - 3 q^{17} - 6 q^{18} - 14 q^{19} - 8 q^{20} - q^{21} + 7 q^{22} + 15 q^{24}+ \cdots - 56 q^{99}+O(q^{100}) $ 7 * q + 2 * q^3 + 8 * q^4 - 7 * q^5 + 2 * q^6 - 3 * q^8 + 13 * q^9 - q^11 - 6 * q^12 + 4 * q^13 - 10 * q^14 - 2 * q^15 - 2 * q^16 - 3 * q^17 - 6 * q^18 - 14 * q^19 - 8 * q^20 - q^21 + 7 * q^22 + 15 * q^24 + 7 * q^25 - 11 * q^26 - 7 * q^27 - 9 * q^28 + 6 * q^29 - 2 * q^30 + 4 * q^32 + 11 * q^33 - 33 * q^34 + 20 * q^36 - 2 * q^38 - 25 * q^39 + 3 * q^40 + 7 * q^41 - 29 * q^42 - 19 * q^43 + 19 * q^44 - 13 * q^45 + 14 * q^46 + 4 * q^47 + 18 * q^48 - 5 * q^49 - 29 * q^51 - 8 * q^52 + 6 * q^53 - 9 * q^54 + q^55 - 25 * q^56 - 44 * q^57 + 12 * q^58 + 2 * q^59 + 6 * q^60 - 4 * q^61 + 25 * q^62 + 5 * q^63 - 31 * q^64 - 4 * q^65 + q^66 + 8 * q^67 - 13 * q^68 - 11 * q^69 + 10 * q^70 - 33 * q^71 + 43 * q^72 + 2 * q^75 - 12 * q^76 + 5 * q^77 + 22 * q^78 - 9 * q^79 + 2 * q^80 + 35 * q^81 - 76 * q^82 + 11 * q^83 - 40 * q^84 + 3 * q^85 - 17 * q^86 - 29 * q^87 - 20 * q^88 - 12 * q^89 + 6 * q^90 - 24 * q^91 - 7 * q^92 - 47 * q^93 + 10 * q^94 + 14 * q^95 - 15 * q^96 - 15 * q^97 + 54 * q^98 - 56 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.12084	−0.792554	−0.396277	−	0.918131i	$-0.629698\pi$
$2$	−1.12084	−0.792554	−0.396277	+	0.918131i	$0.629698\pi$
$3$	3.33999	1.92835	0.964173	−	0.265274i	$-0.0854624\pi$
$3$	3.33999	1.92835	0.964173	+	0.265274i	$0.0854624\pi$
$4$	−0.743718	−0.371859
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−3.74360	−1.52832
$7$	1.22417	0.462694	0.231347	−	0.972871i	$-0.425687\pi$
$7$	1.22417	0.462694	0.231347	+	0.972871i	$0.425687\pi$
$8$	3.07527	1.08727
$9$	8.15556	2.71852
$10$	1.12084	0.354441
$11$	−3.59983	−1.08539	−0.542695	−	0.839930i	$-0.682596\pi$
$11$	−3.59983	−1.08539	−0.542695	+	0.839930i	$0.682596\pi$
$12$	−2.48401	−0.717073
$13$	−0.699609	−0.194037	−0.0970183	−	0.995283i	$-0.530931\pi$
$13$	−0.699609	−0.194037	−0.0970183	+	0.995283i	$0.530931\pi$
$14$	−1.37210	−0.366710
$15$	−3.33999	−0.862383
$16$	−1.95945	−0.489862
$17$	−0.578769	−0.140372	−0.0701861	−	0.997534i	$-0.522359\pi$
$17$	−0.578769	−0.140372	−0.0701861	+	0.997534i	$0.522359\pi$
$18$	−9.14107	−2.15457
$19$	−7.75609	−1.77937	−0.889684	−	0.456577i	$-0.849075\pi$
$19$	−7.75609	−1.77937	−0.889684	+	0.456577i	$0.849075\pi$
$20$	0.743718	0.166300
$21$	4.08873	0.892234
$22$	4.03484	0.860230
$23$	4.47887	0.933910	0.466955	−	0.884281i	$-0.345351\pi$
$23$	4.47887	0.933910	0.466955	+	0.884281i	$0.345351\pi$
$24$	10.2714	2.09664
$25$	1.00000	0.200000
$26$	0.784150	0.153784
$27$	17.2195	3.31390
$28$	−0.910439	−0.172057
$29$	−3.04867	−0.566123	−0.283062	−	0.959102i	$-0.591350\pi$
$29$	−3.04867	−0.566123	−0.283062	+	0.959102i	$0.591350\pi$
$30$	3.74360	0.683484
$31$	−7.98457	−1.43407	−0.717035	−	0.697037i	$-0.754501\pi$
$31$	−7.98457	−1.43407	−0.717035	+	0.697037i	$0.754501\pi$
$32$	−3.95431	−0.699030
$33$	−12.0234	−2.09301
$34$	0.648708	0.111252
$35$	−1.22417	−0.206923
$36$	−6.06543	−1.01091
$37$	0	0
$37$	0	0
$38$	8.69333	1.41024
$39$	−2.33669	−0.374170
$40$	−3.07527	−0.486243
$41$	5.58669	0.872495	0.436247	−	0.899827i	$-0.356307\pi$
$41$	5.58669	0.872495	0.436247	+	0.899827i	$0.356307\pi$
$42$	−4.58281	−0.707143
$43$	−4.02235	−0.613403	−0.306701	−	0.951806i	$-0.599225\pi$
$43$	−4.02235	−0.613403	−0.306701	+	0.951806i	$0.599225\pi$
$44$	2.67726	0.403612
$45$	−8.15556	−1.21576
$46$	−5.02010	−0.740173
$47$	−8.09571	−1.18088	−0.590440	−	0.807081i	$-0.701046\pi$
$47$	−8.09571	−1.18088	−0.590440	+	0.807081i	$0.701046\pi$
$48$	−6.54454	−0.944624
$49$	−5.50140	−0.785914
$50$	−1.12084	−0.158511
$51$	−1.93309	−0.270686
$52$	0.520312	0.0721543
$53$	−9.93959	−1.36531	−0.682654	−	0.730742i	$-0.739174\pi$
$53$	−9.93959	−1.36531	−0.682654	+	0.730742i	$0.739174\pi$
$54$	−19.3003	−2.62644
$55$	3.59983	0.485401
$56$	3.76466	0.503074
$57$	−25.9053	−3.43124
$58$	3.41707	0.448683
$59$	−0.485356	−0.0631879	−0.0315940	−	0.999501i	$-0.510058\pi$
$59$	−0.485356	−0.0631879	−0.0315940	+	0.999501i	$0.510058\pi$
$60$	2.48401	0.320685
$61$	0.919473	0.117726	0.0588632	−	0.998266i	$-0.481252\pi$
$61$	0.919473	0.117726	0.0588632	+	0.998266i	$0.481252\pi$
$62$	8.94942	1.13658
$63$	9.98382	1.25784
$64$	8.35104	1.04388
$65$	0.699609	0.0867758
$66$	13.4763	1.65882
$67$	11.2569	1.37525	0.687624	−	0.726067i	$-0.258654\pi$
$67$	11.2569	1.37525	0.687624	+	0.726067i	$0.258654\pi$
$68$	0.430441	0.0521986
$69$	14.9594	1.80090
$70$	1.37210	0.163998
$71$	−0.507354	−0.0602118	−0.0301059	−	0.999547i	$-0.509584\pi$
$71$	−0.507354	−0.0602118	−0.0301059	+	0.999547i	$0.509584\pi$
$72$	25.0805	2.95577
$73$	−3.55519	−0.416104	−0.208052	−	0.978118i	$-0.566712\pi$
$73$	−3.55519	−0.416104	−0.208052	+	0.978118i	$0.566712\pi$
$74$	0	0
$75$	3.33999	0.385669
$76$	5.76834	0.661674
$77$	−4.40682	−0.502204
$78$	2.61906	0.296550
$79$	3.81414	0.429125	0.214562	−	0.976710i	$-0.431167\pi$
$79$	3.81414	0.429125	0.214562	+	0.976710i	$0.431167\pi$
$80$	1.95945	0.219073
$81$	33.0464	3.67183
$82$	−6.26179	−0.691499
$83$	−11.6109	−1.27446	−0.637230	−	0.770674i	$-0.719920\pi$
$83$	−11.6109	−1.27446	−0.637230	+	0.770674i	$0.719920\pi$
$84$	−3.04086	−0.331785
$85$	0.578769	0.0627763
$86$	4.50841	0.486154
$87$	−10.1825	−1.09168
$88$	−11.0705	−1.18011
$89$	−14.3837	−1.52467	−0.762333	−	0.647185i	$-0.775946\pi$
$89$	−14.3837	−1.52467	−0.762333	+	0.647185i	$0.775946\pi$
$90$	9.14107	0.963554
$91$	−0.856443	−0.0897796
$92$	−3.33102	−0.347283
$93$	−26.6684	−2.76538
$94$	9.07400	0.935911
$95$	7.75609	0.795758
$96$	−13.2074	−1.34797
$97$	10.2579	1.04154	0.520768	−	0.853698i	$-0.325646\pi$
$97$	10.2579	1.04154	0.520768	+	0.853698i	$0.325646\pi$
$98$	6.16619	0.622879
$99$	−29.3586	−2.95065
$100$	−0.743718	−0.0743718
$101$	−5.50183	−0.547452	−0.273726	−	0.961808i	$-0.588256\pi$
$101$	−5.50183	−0.547452	−0.273726	+	0.961808i	$0.588256\pi$
$102$	2.16668	0.214533
$103$	−6.24169	−0.615012	−0.307506	−	0.951546i	$-0.599494\pi$
$103$	−6.24169	−0.615012	−0.307506	+	0.951546i	$0.599494\pi$
$104$	−2.15149	−0.210971
$105$	−4.08873	−0.399019
$106$	11.1407	1.08208
$107$	−0.370138	−0.0357826	−0.0178913	−	0.999840i	$-0.505695\pi$
$107$	−0.370138	−0.0357826	−0.0178913	+	0.999840i	$0.505695\pi$
$108$	−12.8065	−1.23230
$109$	−3.11081	−0.297962	−0.148981	−	0.988840i	$-0.547599\pi$
$109$	−3.11081	−0.297962	−0.148981	+	0.988840i	$0.547599\pi$
$110$	−4.03484	−0.384707
$111$	0	0
$112$	−2.39870	−0.226656
$113$	−11.6924	−1.09992	−0.549962	−	0.835190i	$-0.685358\pi$
$113$	−11.6924	−1.09992	−0.549962	+	0.835190i	$0.685358\pi$
$114$	29.0357	2.71944
$115$	−4.47887	−0.417657
$116$	2.26735	0.210518
$117$	−5.70570	−0.527492
$118$	0.544006	0.0500798
$119$	−0.708514	−0.0649494
$120$	−10.2714	−0.937644
$121$	1.95880	0.178072
$122$	−1.03058	−0.0933045
$123$	18.6595	1.68247
$124$	5.93827	0.533272
$125$	−1.00000	−0.0894427
$126$	−11.1903	−0.996908
$127$	17.7496	1.57503	0.787513	−	0.616298i	$-0.211368\pi$
$127$	17.7496	1.57503	0.787513	+	0.616298i	$0.211368\pi$
$128$	−1.45157	−0.128302
$129$	−13.4346	−1.18285
$130$	−0.784150	−0.0687745
$131$	1.15066	0.100533	0.0502667	−	0.998736i	$-0.483993\pi$
$131$	1.15066	0.100533	0.0502667	+	0.998736i	$0.483993\pi$
$132$	8.94203	0.778304
$133$	−9.49479	−0.823303
$134$	−12.6172	−1.08996
$135$	−17.2195	−1.48202
$136$	−1.77987	−0.152623
$137$	1.91138	0.163301	0.0816503	−	0.996661i	$-0.473981\pi$
$137$	1.91138	0.163301	0.0816503	+	0.996661i	$0.473981\pi$
$138$	−16.7671	−1.42731
$139$	−3.78003	−0.320618	−0.160309	−	0.987067i	$-0.551249\pi$
$139$	−3.78003	−0.320618	−0.160309	+	0.987067i	$0.551249\pi$
$140$	0.910439	0.0769462
$141$	−27.0396	−2.27715
$142$	0.568663	0.0477211
$143$	2.51848	0.210606
$144$	−15.9804	−1.33170
$145$	3.04867	0.253178
$146$	3.98480	0.329784
$147$	−18.3746	−1.51551
$148$	0	0
$149$	−0.552510	−0.0452633	−0.0226317	−	0.999744i	$-0.507205\pi$
$149$	−0.552510	−0.0452633	−0.0226317	+	0.999744i	$0.507205\pi$
$150$	−3.74360	−0.305664
$151$	9.15814	0.745279	0.372639	−	0.927976i	$-0.378453\pi$
$151$	9.15814	0.745279	0.372639	+	0.927976i	$0.378453\pi$
$152$	−23.8520	−1.93466
$153$	−4.72019	−0.381604
$154$	4.93934	0.398023
$155$	7.98457	0.641336
$156$	1.73784	0.139138
$157$	−2.74164	−0.218807	−0.109403	−	0.993997i	$-0.534894\pi$
$157$	−2.74164	−0.218807	−0.109403	+	0.993997i	$0.534894\pi$
$158$	−4.27504	−0.340104
$159$	−33.1982	−2.63279
$160$	3.95431	0.312616
$161$	5.48292	0.432114
$162$	−37.0398	−2.91012
$163$	−21.0689	−1.65025	−0.825123	−	0.564954i	$-0.808894\pi$
$163$	−21.0689	−1.65025	−0.825123	+	0.564954i	$0.808894\pi$
$164$	−4.15492	−0.324445
$165$	12.0234	0.936022
$166$	13.0139	1.01008
$167$	10.0289	0.776063	0.388031	−	0.921646i	$-0.373155\pi$
$167$	10.0289	0.776063	0.388031	+	0.921646i	$0.373155\pi$
$168$	12.5739	0.970101
$169$	−12.5105	−0.962350
$170$	−0.648708	−0.0497536
$171$	−63.2552	−4.83725
$172$	2.99149	0.228099
$173$	−13.3032	−1.01143	−0.505714	−	0.862701i	$-0.668771\pi$
$173$	−13.3032	−1.01143	−0.505714	+	0.862701i	$0.668771\pi$
$174$	11.4130	0.865216
$175$	1.22417	0.0925388
$176$	7.05369	0.531692
$177$	−1.62109	−0.121848
$178$	16.1218	1.20838
$179$	−9.15637	−0.684379	−0.342190	−	0.939631i	$-0.611168\pi$
$179$	−9.15637	−0.684379	−0.342190	+	0.939631i	$0.611168\pi$
$180$	6.06543	0.452091
$181$	21.1056	1.56876	0.784382	−	0.620278i	$-0.212980\pi$
$181$	21.1056	1.56876	0.784382	+	0.620278i	$0.212980\pi$
$182$	0.959935	0.0711551
$183$	3.07103	0.227017
$184$	13.7737	1.01541
$185$	0	0
$186$	29.8910	2.19172
$187$	2.08347	0.152359
$188$	6.02092	0.439121
$189$	21.0797	1.53332
$190$	−8.69333	−0.630680
$191$	7.51240	0.543578	0.271789	−	0.962357i	$-0.412385\pi$
$191$	7.51240	0.543578	0.271789	+	0.962357i	$0.412385\pi$
$192$	27.8924	2.01296
$193$	−16.5435	−1.19083	−0.595415	−	0.803418i	$-0.703012\pi$
$193$	−16.5435	−1.19083	−0.595415	+	0.803418i	$0.703012\pi$
$194$	−11.4975	−0.825473
$195$	2.33669	0.167334
$196$	4.09149	0.292249
$197$	1.74676	0.124452	0.0622259	−	0.998062i	$-0.480180\pi$
$197$	1.74676	0.124452	0.0622259	+	0.998062i	$0.480180\pi$
$198$	32.9063	2.33855
$199$	14.8223	1.05073	0.525364	−	0.850878i	$-0.323929\pi$
$199$	14.8223	1.05073	0.525364	+	0.850878i	$0.323929\pi$
$200$	3.07527	0.217454
$201$	37.5980	2.65196
$202$	6.16667	0.433885
$203$	−3.73210	−0.261942
$204$	1.43767	0.100657
$205$	−5.58669	−0.390192
$206$	6.99594	0.487430
$207$	36.5277	2.53885
$208$	1.37085	0.0950512
$209$	27.9206	1.93131
$210$	4.58281	0.316244
$211$	−18.1359	−1.24853	−0.624264	−	0.781214i	$-0.714601\pi$
$211$	−18.1359	−1.24853	−0.624264	+	0.781214i	$0.714601\pi$
$212$	7.39225	0.507702
$213$	−1.69456	−0.116109
$214$	0.414866	0.0283596
$215$	4.02235	0.274322
$216$	52.9547	3.60311
$217$	−9.77450	−0.663536
$218$	3.48673	0.236151
$219$	−11.8743	−0.802392
$220$	−2.67726	−0.180501
$221$	0.404912	0.0272373
$222$	0	0
$223$	−3.84578	−0.257532	−0.128766	−	0.991675i	$-0.541102\pi$
$223$	−3.84578	−0.257532	−0.128766	+	0.991675i	$0.541102\pi$
$224$	−4.84076	−0.323437
$225$	8.15556	0.543704
$226$	13.1053	0.871749
$227$	13.8135	0.916835	0.458418	−	0.888737i	$-0.348416\pi$
$227$	13.8135	0.916835	0.458418	+	0.888737i	$0.348416\pi$
$228$	19.2662	1.27594
$229$	0.602997	0.0398472	0.0199236	−	0.999802i	$-0.493658\pi$
$229$	0.602997	0.0398472	0.0199236	+	0.999802i	$0.493658\pi$
$230$	5.02010	0.331016
$231$	−14.7187	−0.968423
$232$	−9.37547	−0.615530
$233$	23.3424	1.52921	0.764605	−	0.644499i	$-0.222934\pi$
$233$	23.3424	1.52921	0.764605	+	0.644499i	$0.222934\pi$
$234$	6.39518	0.418066
$235$	8.09571	0.528106
$236$	0.360968	0.0234970
$237$	12.7392	0.827501
$238$	0.794131	0.0514758
$239$	28.0935	1.81722	0.908610	−	0.417645i	$-0.137145\pi$
$239$	28.0935	1.81722	0.908610	+	0.417645i	$0.137145\pi$
$240$	6.54454	0.422449
$241$	−0.0364043	−0.00234501	−0.00117250	−	0.999999i	$-0.500373\pi$
$241$	−0.0364043	−0.00234501	−0.00117250	+	0.999999i	$0.500373\pi$
$242$	−2.19550	−0.141132
$243$	58.7163	3.76665
$244$	−0.683828	−0.0437776
$245$	5.50140	0.351472
$246$	−20.9143	−1.33345
$247$	5.42623	0.345263
$248$	−24.5547	−1.55922
$249$	−38.7803	−2.45760
$250$	1.12084	0.0708881
$251$	21.4173	1.35185	0.675924	−	0.736971i	$-0.263745\pi$
$251$	21.4173	1.35185	0.675924	+	0.736971i	$0.263745\pi$
$252$	−7.42514	−0.467740
$253$	−16.1232	−1.01366
$254$	−19.8945	−1.24829
$255$	1.93309	0.121055
$256$	−15.0751	−0.942195
$257$	0.807757	0.0503865	0.0251933	−	0.999683i	$-0.491980\pi$
$257$	0.807757	0.0503865	0.0251933	+	0.999683i	$0.491980\pi$
$258$	15.0581	0.937474
$259$	0	0
$260$	−0.520312	−0.0322684
$261$	−24.8636	−1.53902
$262$	−1.28970	−0.0796780
$263$	2.43866	0.150374	0.0751871	−	0.997169i	$-0.476045\pi$
$263$	2.43866	0.150374	0.0751871	+	0.997169i	$0.476045\pi$
$264$	−36.9752	−2.27567
$265$	9.93959	0.610584
$266$	10.6421	0.652512
$267$	−48.0414	−2.94008
$268$	−8.37195	−0.511398
$269$	12.1567	0.741208	0.370604	−	0.928791i	$-0.379151\pi$
$269$	12.1567	0.741208	0.370604	+	0.928791i	$0.379151\pi$
$270$	19.3003	1.17458
$271$	−31.3028	−1.90151	−0.950755	−	0.309945i	$-0.899690\pi$
$271$	−31.3028	−1.90151	−0.950755	+	0.309945i	$0.899690\pi$
$272$	1.13407	0.0687630
$273$	−2.86051	−0.173126
$274$	−2.14236	−0.129424
$275$	−3.59983	−0.217078
$276$	−11.1256	−0.669681
$277$	−23.1919	−1.39346	−0.696732	−	0.717331i	$-0.745363\pi$
$277$	−23.1919	−1.39346	−0.696732	+	0.717331i	$0.745363\pi$
$278$	4.23681	0.254107
$279$	−65.1186	−3.89855
$280$	−3.76466	−0.224982
$281$	21.4228	1.27798	0.638988	−	0.769217i	$-0.279353\pi$
$281$	21.4228	1.27798	0.638988	+	0.769217i	$0.279353\pi$
$282$	30.3071	1.80476
$283$	13.8157	0.821257	0.410628	−	0.911803i	$-0.365309\pi$
$283$	13.8157	0.821257	0.410628	+	0.911803i	$0.365309\pi$
$284$	0.377328	0.0223903
$285$	25.9053	1.53450
$286$	−2.82281	−0.166916
$287$	6.83908	0.403698
$288$	−32.2496	−1.90033
$289$	−16.6650	−0.980296
$290$	−3.41707	−0.200657
$291$	34.2615	2.00844
$292$	2.64406	0.154732
$293$	−7.10739	−0.415218	−0.207609	−	0.978212i	$-0.566568\pi$
$293$	−7.10739	−0.415218	−0.207609	+	0.978212i	$0.566568\pi$
$294$	20.5950	1.20113
$295$	0.485356	0.0282585
$296$	0	0
$297$	−61.9874	−3.59687
$298$	0.619275	0.0358736
$299$	−3.13346	−0.181213
$300$	−2.48401	−0.143415
$301$	−4.92405	−0.283818
$302$	−10.2648	−0.590673
$303$	−18.3761	−1.05568
$304$	15.1976	0.871645
$305$	−0.919473	−0.0526489
$306$	5.29057	0.302442
$307$	−23.6385	−1.34912	−0.674560	−	0.738220i	$-0.735667\pi$
$307$	−23.6385	−1.34912	−0.674560	+	0.738220i	$0.735667\pi$
$308$	3.27743	0.186749
$309$	−20.8472	−1.18596
$310$	−8.94942	−0.508293
$311$	19.3609	1.09785	0.548927	−	0.835870i	$-0.315037\pi$
$311$	19.3609	1.09785	0.548927	+	0.835870i	$0.315037\pi$
$312$	−7.18595	−0.406824
$313$	−6.11046	−0.345384	−0.172692	−	0.984976i	$-0.555246\pi$
$313$	−6.11046	−0.345384	−0.172692	+	0.984976i	$0.555246\pi$
$314$	3.07294	0.173416
$315$	−9.98382	−0.562524
$316$	−2.83665	−0.159574
$317$	−33.2406	−1.86698	−0.933490	−	0.358603i	$-0.883253\pi$
$317$	−33.2406	−1.86698	−0.933490	+	0.358603i	$0.883253\pi$
$318$	37.2098	2.08662
$319$	10.9747	0.614465
$320$	−8.35104	−0.466838
$321$	−1.23626	−0.0690012
$322$	−6.14547	−0.342474
$323$	4.48898	0.249774
$324$	−24.5772	−1.36540
$325$	−0.699609	−0.0388073
$326$	23.6149	1.30791
$327$	−10.3901	−0.574574
$328$	17.1806	0.948639
$329$	−9.91055	−0.546387
$330$	−13.4763	−0.741847
$331$	−1.38735	−0.0762557	−0.0381278	−	0.999273i	$-0.512139\pi$
$331$	−1.38735	−0.0762557	−0.0381278	+	0.999273i	$0.512139\pi$
$332$	8.63522	0.473919
$333$	0	0
$334$	−11.2408	−0.615071
$335$	−11.2569	−0.615030
$336$	−8.01166	−0.437072
$337$	15.3098	0.833977	0.416989	−	0.908912i	$-0.363086\pi$
$337$	15.3098	0.833977	0.416989	+	0.908912i	$0.363086\pi$
$338$	14.0223	0.762714
$339$	−39.0524	−2.12103
$340$	−0.430441	−0.0233439
$341$	28.7431	1.55653
$342$	70.8989	3.83378
$343$	−15.3039	−0.826332
$344$	−12.3698	−0.666935
$345$	−14.9594	−0.805387
$346$	14.9108	0.801610
$347$	−27.0099	−1.44997	−0.724983	−	0.688767i	$-0.758152\pi$
$347$	−27.0099	−1.44997	−0.724983	+	0.688767i	$0.758152\pi$
$348$	7.57293	0.405951
$349$	−20.3183	−1.08761	−0.543806	−	0.839211i	$-0.683017\pi$
$349$	−20.3183	−1.08761	−0.543806	+	0.839211i	$0.683017\pi$
$350$	−1.37210	−0.0733420
$351$	−12.0469	−0.643018
$352$	14.2349	0.758720
$353$	4.47020	0.237925	0.118962	−	0.992899i	$-0.462043\pi$
$353$	4.47020	0.237925	0.118962	+	0.992899i	$0.462043\pi$
$354$	1.81698	0.0965713
$355$	0.507354	0.0269275
$356$	10.6974	0.566960
$357$	−2.36643	−0.125245
$358$	10.2628	0.542407
$359$	−31.2742	−1.65059	−0.825295	−	0.564702i	$-0.808991\pi$
$359$	−31.2742	−1.65059	−0.825295	+	0.564702i	$0.808991\pi$
$360$	−25.0805	−1.32186
$361$	41.1569	2.16615
$362$	−23.6560	−1.24333
$363$	6.54236	0.343385
$364$	0.636952	0.0333853
$365$	3.55519	0.186087
$366$	−3.44214	−0.179923
$367$	−16.1265	−0.841796	−0.420898	−	0.907108i	$-0.638285\pi$
$367$	−16.1265	−0.841796	−0.420898	+	0.907108i	$0.638285\pi$
$368$	−8.77612	−0.457487
$369$	45.5626	2.37189
$370$	0	0
$371$	−12.1678	−0.631720
$372$	19.8338	1.02833
$373$	−10.2403	−0.530224	−0.265112	−	0.964218i	$-0.585409\pi$
$373$	−10.2403	−0.530224	−0.265112	+	0.964218i	$0.585409\pi$
$374$	−2.33524	−0.120752
$375$	−3.33999	−0.172477
$376$	−24.8965	−1.28394
$377$	2.13288	0.109849
$378$	−23.6270	−1.21524
$379$	16.5621	0.850740	0.425370	−	0.905019i	$-0.360144\pi$
$379$	16.5621	0.850740	0.425370	+	0.905019i	$0.360144\pi$
$380$	−5.76834	−0.295910
$381$	59.2837	3.03720
$382$	−8.42020	−0.430815
$383$	−18.3014	−0.935157	−0.467578	−	0.883952i	$-0.654873\pi$
$383$	−18.3014	−0.935157	−0.467578	+	0.883952i	$0.654873\pi$
$384$	−4.84822	−0.247410
$385$	4.40682	0.224592
$386$	18.5427	0.943796
$387$	−32.8045	−1.66755
$388$	−7.62901	−0.387304
$389$	14.6722	0.743908	0.371954	−	0.928251i	$-0.378688\pi$
$389$	14.6722	0.743908	0.371954	+	0.928251i	$0.378688\pi$
$390$	−2.61906	−0.132621
$391$	−2.59223	−0.131095
$392$	−16.9183	−0.854502
$393$	3.84319	0.193863
$394$	−1.95784	−0.0986346
$395$	−3.81414	−0.191910
$396$	21.8345	1.09723
$397$	−2.00229	−0.100492	−0.0502460	−	0.998737i	$-0.516001\pi$
$397$	−2.00229	−0.100492	−0.0502460	+	0.998737i	$0.516001\pi$
$398$	−16.6135	−0.832757
$399$	−31.7125	−1.58761
$400$	−1.95945	−0.0979724
$401$	22.8854	1.14284	0.571422	−	0.820656i	$-0.306392\pi$
$401$	22.8854	1.14284	0.571422	+	0.820656i	$0.306392\pi$
$402$	−42.1413	−2.10182
$403$	5.58608	0.278262
$404$	4.09181	0.203575
$405$	−33.0464	−1.64209
$406$	4.18308	0.207603
$407$	0	0
$408$	−5.94476	−0.294309
$409$	−9.11629	−0.450771	−0.225386	−	0.974270i	$-0.572364\pi$
$409$	−9.11629	−0.450771	−0.225386	+	0.974270i	$0.572364\pi$
$410$	6.26179	0.309248
$411$	6.38401	0.314900
$412$	4.64206	0.228698
$413$	−0.594160	−0.0292367
$414$	−40.9417	−2.01218
$415$	11.6109	0.569956
$416$	2.76647	0.135637
$417$	−12.6253	−0.618263
$418$	−31.2945	−1.53067
$419$	−8.51059	−0.415770	−0.207885	−	0.978153i	$-0.566658\pi$
$419$	−8.51059	−0.415770	−0.207885	+	0.978153i	$0.566658\pi$
$420$	3.04086	0.148379
$421$	8.83588	0.430635	0.215317	−	0.976544i	$-0.430921\pi$
$421$	8.83588	0.430635	0.215317	+	0.976544i	$0.430921\pi$
$422$	20.3274	0.989525
$423$	−66.0250	−3.21025
$424$	−30.5669	−1.48446
$425$	−0.578769	−0.0280744
$426$	1.89933	0.0920228
$427$	1.12559	0.0544713
$428$	0.275278	0.0133061
$429$	8.41169	0.406120
$430$	−4.50841	−0.217415
$431$	−1.51065	−0.0727656	−0.0363828	−	0.999338i	$-0.511584\pi$
$431$	−1.51065	−0.0727656	−0.0363828	+	0.999338i	$0.511584\pi$
$432$	−33.7408	−1.62335
$433$	6.85824	0.329586	0.164793	−	0.986328i	$-0.447304\pi$
$433$	6.85824	0.329586	0.164793	+	0.986328i	$0.447304\pi$
$434$	10.9556	0.525888
$435$	10.1825	0.488215
$436$	2.31357	0.110800
$437$	−34.7385	−1.66177
$438$	13.3092	0.635938
$439$	−19.1378	−0.913398	−0.456699	−	0.889621i	$-0.650968\pi$
$439$	−19.1378	−0.913398	−0.456699	+	0.889621i	$0.650968\pi$
$440$	11.0705	0.527763
$441$	−44.8670	−2.13652
$442$	−0.453842	−0.0215871
$443$	−2.25690	−0.107229	−0.0536144	−	0.998562i	$-0.517074\pi$
$443$	−2.25690	−0.107229	−0.0536144	+	0.998562i	$0.517074\pi$
$444$	0	0
$445$	14.3837	0.681851
$446$	4.31050	0.204108
$447$	−1.84538	−0.0872834
$448$	10.2231	0.482997
$449$	24.3779	1.15046	0.575232	−	0.817991i	$-0.304912\pi$
$449$	24.3779	1.15046	0.575232	+	0.817991i	$0.304912\pi$
$450$	−9.14107	−0.430914
$451$	−20.1112	−0.946998
$452$	8.69581	0.409017
$453$	30.5881	1.43716
$454$	−15.4827	−0.726641
$455$	0.856443	0.0401507
$456$	−79.6657	−3.73069
$457$	14.0526	0.657352	0.328676	−	0.944443i	$-0.393398\pi$
$457$	14.0526	0.657352	0.328676	+	0.944443i	$0.393398\pi$
$458$	−0.675864	−0.0315810
$459$	−9.96613	−0.465179
$460$	3.33102	0.155309
$461$	26.4176	1.23039	0.615195	−	0.788375i	$-0.289077\pi$
$461$	26.4176	1.23039	0.615195	+	0.788375i	$0.289077\pi$
$462$	16.4974	0.767527
$463$	−25.1424	−1.16847	−0.584233	−	0.811586i	$-0.698604\pi$
$463$	−25.1424	−1.16847	−0.584233	+	0.811586i	$0.698604\pi$
$464$	5.97370	0.277322
$465$	26.6684	1.23672
$466$	−26.1631	−1.21198
$467$	14.0309	0.649271	0.324635	−	0.945839i	$-0.394758\pi$
$467$	14.0309	0.649271	0.324635	+	0.945839i	$0.394758\pi$
$468$	4.24343	0.196153
$469$	13.7804	0.636319
$470$	−9.07400	−0.418552
$471$	−9.15705	−0.421935
$472$	−1.49260	−0.0687025
$473$	14.4798	0.665781
$474$	−14.2786	−0.655839
$475$	−7.75609	−0.355874
$476$	0.526934	0.0241520
$477$	−81.0629	−3.71161
$478$	−31.4884	−1.44024
$479$	6.68212	0.305314	0.152657	−	0.988279i	$-0.451217\pi$
$479$	6.68212	0.305314	0.152657	+	0.988279i	$0.451217\pi$
$480$	13.2074	0.602831
$481$	0	0
$482$	0.0408034	0.00185854
$483$	18.3129	0.833266
$484$	−1.45679	−0.0662178
$485$	−10.2579	−0.465789
$486$	−65.8116	−2.98527
$487$	7.26544	0.329229	0.164614	−	0.986358i	$-0.447362\pi$
$487$	7.26544	0.329229	0.164614	+	0.986358i	$0.447362\pi$
$488$	2.82763	0.128001
$489$	−70.3701	−3.18224
$490$	−6.16619	−0.278560
$491$	0.446623	0.0201558	0.0100779	−	0.999949i	$-0.496792\pi$
$491$	0.446623	0.0201558	0.0100779	+	0.999949i	$0.496792\pi$
$492$	−13.8774	−0.625642
$493$	1.76447	0.0794679
$494$	−6.08193	−0.273639
$495$	29.3586	1.31957
$496$	15.6453	0.702497
$497$	−0.621089	−0.0278597
$498$	43.4665	1.94778
$499$	10.6045	0.474722	0.237361	−	0.971421i	$-0.423718\pi$
$499$	10.6045	0.474722	0.237361	+	0.971421i	$0.423718\pi$
$500$	0.743718	0.0332601
$501$	33.4966	1.49652
$502$	−24.0054	−1.07141
$503$	−13.9696	−0.622872	−0.311436	−	0.950267i	$-0.600810\pi$
$503$	−13.9696	−0.622872	−0.311436	+	0.950267i	$0.600810\pi$
$504$	30.7029	1.36762
$505$	5.50183	0.244828
$506$	18.0715	0.803377
$507$	−41.7851	−1.85574
$508$	−13.2007	−0.585687
$509$	39.6334	1.75672	0.878360	−	0.478000i	$-0.158638\pi$
$509$	39.6334	1.75672	0.878360	+	0.478000i	$0.158638\pi$
$510$	−2.16668	−0.0959422
$511$	−4.35217	−0.192529
$512$	19.7999	0.875041
$513$	−133.556	−5.89665
$514$	−0.905366	−0.0399340
$515$	6.24169	0.275042
$516$	9.99157	0.439854
$517$	29.1432	1.28172
$518$	0	0
$519$	−44.4328	−1.95038
$520$	2.15149	0.0943489
$521$	−31.4296	−1.37695	−0.688477	−	0.725258i	$-0.741720\pi$
$521$	−31.4296	−1.37695	−0.688477	+	0.725258i	$0.741720\pi$
$522$	27.8681	1.21975
$523$	−14.3494	−0.627458	−0.313729	−	0.949513i	$-0.601578\pi$
$523$	−14.3494	−0.627458	−0.313729	+	0.949513i	$0.601578\pi$
$524$	−0.855764	−0.0373842
$525$	4.08873	0.178447
$526$	−2.73335	−0.119180
$527$	4.62122	0.201304
$528$	23.5593	1.02529
$529$	−2.93969	−0.127813
$530$	−11.1407	−0.483921
$531$	−3.95835	−0.171778
$532$	7.06145	0.306153
$533$	−3.90850	−0.169296
$534$	53.8467	2.33017
$535$	0.370138	0.0160025
$536$	34.6180	1.49527
$537$	−30.5822	−1.31972
$538$	−13.6257	−0.587447
$539$	19.8041	0.853024
$540$	12.8065	0.551103
$541$	−14.4219	−0.620046	−0.310023	−	0.950729i	$-0.600337\pi$
$541$	−14.4219	−0.620046	−0.310023	+	0.950729i	$0.600337\pi$
$542$	35.0854	1.50705
$543$	70.4924	3.02512
$544$	2.28863	0.0981243
$545$	3.11081	0.133253
$546$	3.20618	0.137212
$547$	−4.85582	−0.207620	−0.103810	−	0.994597i	$-0.533103\pi$
$547$	−4.85582	−0.207620	−0.103810	+	0.994597i	$0.533103\pi$
$548$	−1.42153	−0.0607247
$549$	7.49881	0.320041
$550$	4.03484	0.172046
$551$	23.6457	1.00734
$552$	46.0042	1.95807
$553$	4.66917	0.198553
$554$	25.9944	1.10439
$555$	0	0
$556$	2.81128	0.119225
$557$	4.74819	0.201187	0.100594	−	0.994928i	$-0.467926\pi$
$557$	4.74819	0.201187	0.100594	+	0.994928i	$0.467926\pi$
$558$	72.9875	3.08981
$559$	2.81407	0.119023
$560$	2.39870	0.101364
$561$	6.95878	0.293800
$562$	−24.0115	−1.01286
$563$	−12.4349	−0.524069	−0.262035	−	0.965058i	$-0.584393\pi$
$563$	−12.4349	−0.524069	−0.262035	+	0.965058i	$0.584393\pi$
$564$	20.1098	0.846777
$565$	11.6924	0.491901
$566$	−15.4852	−0.650890
$567$	40.4546	1.69893
$568$	−1.56025	−0.0654666
$569$	−11.9189	−0.499666	−0.249833	−	0.968289i	$-0.580376\pi$
$569$	−11.9189	−0.499666	−0.249833	+	0.968289i	$0.580376\pi$
$570$	−29.0357	−1.21617
$571$	−27.4093	−1.14704	−0.573522	−	0.819190i	$-0.694423\pi$
$571$	−27.4093	−1.14704	−0.573522	+	0.819190i	$0.694423\pi$
$572$	−1.87304	−0.0783155
$573$	25.0914	1.04821
$574$	−7.66552	−0.319952
$575$	4.47887	0.186782
$576$	68.1074	2.83781
$577$	6.47915	0.269731	0.134865	−	0.990864i	$-0.456940\pi$
$577$	6.47915	0.269731	0.134865	+	0.990864i	$0.456940\pi$
$578$	18.6788	0.776937
$579$	−55.2553	−2.29633
$580$	−2.26735	−0.0941465
$581$	−14.2137	−0.589685
$582$	−38.4016	−1.59180
$583$	35.7809	1.48189
$584$	−10.9332	−0.452418
$585$	5.70570	0.235902
$586$	7.96625	0.329083
$587$	−38.8629	−1.60404	−0.802022	−	0.597295i	$-0.796242\pi$
$587$	−38.8629	−1.60404	−0.802022	+	0.597295i	$0.796242\pi$
$588$	13.6655	0.563558
$589$	61.9290	2.55174
$590$	−0.544006	−0.0223964
$591$	5.83418	0.239986
$592$	0	0
$593$	35.6076	1.46223	0.731115	−	0.682254i	$-0.239000\pi$
$593$	35.6076	1.46223	0.731115	+	0.682254i	$0.239000\pi$
$594$	69.4780	2.85072
$595$	0.708514	0.0290462
$596$	0.410911	0.0168316
$597$	49.5065	2.02617
$598$	3.51211	0.143621
$599$	16.9233	0.691469	0.345735	−	0.938332i	$-0.387630\pi$
$599$	16.9233	0.691469	0.345735	+	0.938332i	$0.387630\pi$
$600$	10.2714	0.419327
$601$	−13.2569	−0.540760	−0.270380	−	0.962754i	$-0.587149\pi$
$601$	−13.2569	−0.540760	−0.270380	+	0.962754i	$0.587149\pi$
$602$	5.51908	0.224941
$603$	91.8062	3.73864
$604$	−6.81107	−0.277139
$605$	−1.95880	−0.0796364
$606$	20.5966	0.836681
$607$	5.76173	0.233861	0.116931	−	0.993140i	$-0.462694\pi$
$607$	5.76173	0.233861	0.116931	+	0.993140i	$0.462694\pi$
$608$	30.6700	1.24383
$609$	−12.4652	−0.505114
$610$	1.03058	0.0417270
$611$	5.66383	0.229134
$612$	3.51049	0.141903
$613$	−33.3853	−1.34842	−0.674210	−	0.738540i	$-0.735516\pi$
$613$	−33.3853	−1.34842	−0.674210	+	0.738540i	$0.735516\pi$
$614$	26.4950	1.06925
$615$	−18.6595	−0.752424
$616$	−13.5522	−0.546032
$617$	−33.6754	−1.35572	−0.677860	−	0.735191i	$-0.737092\pi$
$617$	−33.6754	−1.35572	−0.677860	+	0.735191i	$0.737092\pi$
$618$	23.3664	0.939934
$619$	28.0530	1.12755	0.563774	−	0.825929i	$-0.309349\pi$
$619$	28.0530	1.12755	0.563774	+	0.825929i	$0.309349\pi$
$620$	−5.93827	−0.238486
$621$	77.1241	3.09488
$622$	−21.7004	−0.870108
$623$	−17.6081	−0.705454
$624$	4.57862	0.183292
$625$	1.00000	0.0400000
$626$	6.84885	0.273735
$627$	93.2547	3.72423
$628$	2.03900	0.0813652
$629$	0	0
$630$	11.1903	0.445831
$631$	−31.6599	−1.26036	−0.630181	−	0.776448i	$-0.717019\pi$
$631$	−31.6599	−1.26036	−0.630181	+	0.776448i	$0.717019\pi$
$632$	11.7295	0.466575
$633$	−60.5738	−2.40759
$634$	37.2574	1.47968
$635$	−17.7496	−0.704373
$636$	24.6901	0.979025
$637$	3.84883	0.152496
$638$	−12.3009	−0.486996
$639$	−4.13775	−0.163687
$640$	1.45157	0.0573782
$641$	−2.08696	−0.0824299	−0.0412149	−	0.999150i	$-0.513123\pi$
$641$	−2.08696	−0.0824299	−0.0412149	+	0.999150i	$0.513123\pi$
$642$	1.38565	0.0546872
$643$	−1.07099	−0.0422359	−0.0211179	−	0.999777i	$-0.506723\pi$
$643$	−1.07099	−0.0422359	−0.0211179	+	0.999777i	$0.506723\pi$
$644$	−4.07774	−0.160686
$645$	13.4346	0.528988
$646$	−5.03143	−0.197959
$647$	34.8927	1.37177	0.685886	−	0.727709i	$-0.259415\pi$
$647$	34.8927	1.37177	0.685886	+	0.727709i	$0.259415\pi$
$648$	101.627	3.99227
$649$	1.74720	0.0685836
$650$	0.784150	0.0307569
$651$	−32.6468	−1.27953
$652$	15.6693	0.613658
$653$	1.96349	0.0768372	0.0384186	−	0.999262i	$-0.487768\pi$
$653$	1.96349	0.0768372	0.0384186	+	0.999262i	$0.487768\pi$
$654$	11.6456	0.455381
$655$	−1.15066	−0.0449599
$656$	−10.9468	−0.427402
$657$	−28.9946	−1.13119
$658$	11.1081	0.433041
$659$	19.5562	0.761802	0.380901	−	0.924616i	$-0.375614\pi$
$659$	19.5562	0.761802	0.380901	+	0.924616i	$0.375614\pi$
$660$	−8.94203	−0.348068
$661$	−1.97420	−0.0767873	−0.0383937	−	0.999263i	$-0.512224\pi$
$661$	−1.97420	−0.0767873	−0.0383937	+	0.999263i	$0.512224\pi$
$662$	1.55500	0.0604367
$663$	1.35240	0.0525230
$664$	−35.7066	−1.38568
$665$	9.49479	0.368192
$666$	0	0
$667$	−13.6546	−0.528708
$668$	−7.45870	−0.288586
$669$	−12.8449	−0.496611
$670$	12.6172	0.487444
$671$	−3.30995	−0.127779
$672$	−16.1681	−0.623698
$673$	20.6652	0.796583	0.398292	−	0.917259i	$-0.369603\pi$
$673$	20.6652	0.796583	0.398292	+	0.917259i	$0.369603\pi$
$674$	−17.1598	−0.660971
$675$	17.2195	0.662780
$676$	9.30432	0.357858
$677$	−10.7458	−0.412994	−0.206497	−	0.978447i	$-0.566206\pi$
$677$	−10.7458	−0.412994	−0.206497	+	0.978447i	$0.566206\pi$
$678$	43.7715	1.68103
$679$	12.5575	0.481913
$680$	1.77987	0.0682549
$681$	46.1371	1.76798
$682$	−32.2164	−1.23363
$683$	−12.0523	−0.461170	−0.230585	−	0.973052i	$-0.574064\pi$
$683$	−12.0523	−0.461170	−0.230585	+	0.973052i	$0.574064\pi$
$684$	47.0440	1.79877
$685$	−1.91138	−0.0730302
$686$	17.1532	0.654912
$687$	2.01401	0.0768392
$688$	7.88159	0.300483
$689$	6.95383	0.264920
$690$	16.7671	0.638313
$691$	10.3135	0.392344	0.196172	−	0.980570i	$-0.437149\pi$
$691$	10.3135	0.392344	0.196172	+	0.980570i	$0.437149\pi$
$692$	9.89386	0.376108
$693$	−35.9401	−1.36525
$694$	30.2737	1.14917
$695$	3.78003	0.143385
$696$	−31.3140	−1.18695
$697$	−3.23341	−0.122474
$698$	22.7735	0.861991
$699$	77.9634	2.94885
$700$	−0.910439	−0.0344114
$701$	−35.5154	−1.34140	−0.670699	−	0.741729i	$-0.734006\pi$
$701$	−35.5154	−1.34140	−0.670699	+	0.741729i	$0.734006\pi$
$702$	13.5027	0.509626
$703$	0	0
$704$	−30.0624	−1.13302
$705$	27.0396	1.01837
$706$	−5.01038	−0.188568
$707$	−6.73519	−0.253303
$708$	1.20563	0.0453104
$709$	32.6695	1.22693	0.613465	−	0.789722i	$-0.289775\pi$
$709$	32.6695	1.22693	0.613465	+	0.789722i	$0.289775\pi$
$710$	−0.568663	−0.0213415
$711$	31.1065	1.16658
$712$	−44.2336	−1.65773
$713$	−35.7619	−1.33929
$714$	2.65239	0.0992632
$715$	−2.51848	−0.0941857
$716$	6.80975	0.254492
$717$	93.8323	3.50423
$718$	35.0534	1.30818
$719$	32.6486	1.21759	0.608794	−	0.793328i	$-0.291653\pi$
$719$	32.6486	1.21759	0.608794	+	0.793328i	$0.291653\pi$
$720$	15.9804	0.595554
$721$	−7.64092	−0.284563
$722$	−46.1302	−1.71679
$723$	−0.121590	−0.00452198
$724$	−15.6966	−0.583359
$725$	−3.04867	−0.113225
$726$	−7.33294	−0.272151
$727$	14.1820	0.525980	0.262990	−	0.964799i	$-0.415291\pi$
$727$	14.1820	0.525980	0.262990	+	0.964799i	$0.415291\pi$
$728$	−2.63379	−0.0976148
$729$	96.9728	3.59158
$730$	−3.98480	−0.147484
$731$	2.32801	0.0861046
$732$	−2.28398	−0.0844184
$733$	19.4533	0.718526	0.359263	−	0.933236i	$-0.383028\pi$
$733$	19.4533	0.718526	0.359263	+	0.933236i	$0.383028\pi$
$734$	18.0752	0.667168
$735$	18.3746	0.677759
$736$	−17.7108	−0.652831
$737$	−40.5229	−1.49268
$738$	−51.0684	−1.87985
$739$	−52.1774	−1.91938	−0.959689	−	0.281063i	$-0.909313\pi$
$739$	−52.1774	−1.91938	−0.959689	+	0.281063i	$0.909313\pi$
$740$	0	0
$741$	18.1236	0.665786
$742$	13.6381	0.500672
$743$	13.3673	0.490397	0.245198	−	0.969473i	$-0.421147\pi$
$743$	13.3673	0.490397	0.245198	+	0.969473i	$0.421147\pi$
$744$	−82.0125	−3.00672
$745$	0.552510	0.0202424
$746$	11.4778	0.420231
$747$	−94.6932	−3.46464
$748$	−1.54952	−0.0566559
$749$	−0.453113	−0.0165564
$750$	3.74360	0.136697
$751$	16.7410	0.610887	0.305443	−	0.952210i	$-0.401195\pi$
$751$	16.7410	0.610887	0.305443	+	0.952210i	$0.401195\pi$
$752$	15.8631	0.578469
$753$	71.5337	2.60683
$754$	−2.39061	−0.0870609
$755$	−9.15814	−0.333299
$756$	−15.6773	−0.570179
$757$	11.2787	0.409932	0.204966	−	0.978769i	$-0.434292\pi$
$757$	11.2787	0.409932	0.204966	+	0.978769i	$0.434292\pi$
$758$	−18.5635	−0.674257
$759$	−53.8514	−1.95468
$760$	23.8520	0.865205
$761$	−46.7462	−1.69455	−0.847275	−	0.531155i	$-0.821758\pi$
$761$	−46.7462	−1.69455	−0.847275	+	0.531155i	$0.821758\pi$
$762$	−66.4475	−2.40714
$763$	−3.80818	−0.137865
$764$	−5.58711	−0.202134
$765$	4.72019	0.170659
$766$	20.5129	0.741162
$767$	0.339559	0.0122608
$768$	−50.3508	−1.81688
$769$	−37.8487	−1.36486	−0.682430	−	0.730951i	$-0.739077\pi$
$769$	−37.8487	−1.36486	−0.682430	+	0.730951i	$0.739077\pi$
$770$	−4.93934	−0.178001
$771$	2.69790	0.0971626
$772$	12.3037	0.442821
$773$	−4.14880	−0.149222	−0.0746110	−	0.997213i	$-0.523772\pi$
$773$	−4.14880	−0.149222	−0.0746110	+	0.997213i	$0.523772\pi$
$774$	36.7686	1.32162
$775$	−7.98457	−0.286814
$776$	31.5459	1.13243
$777$	0	0
$778$	−16.4451	−0.589587
$779$	−43.3309	−1.55249
$780$	−1.73784	−0.0622246
$781$	1.82639	0.0653533
$782$	2.90548	0.103900
$783$	−52.4966	−1.87608
$784$	10.7797	0.384990
$785$	2.74164	0.0978533
$786$	−4.30760	−0.153647
$787$	−42.8205	−1.52639	−0.763193	−	0.646170i	$-0.776370\pi$
$787$	−42.8205	−1.52639	−0.763193	+	0.646170i	$0.776370\pi$
$788$	−1.29910	−0.0462785
$789$	8.14511	0.289974
$790$	4.27504	0.152099
$791$	−14.3135	−0.508928
$792$	−90.2857	−3.20816
$793$	−0.643271	−0.0228432
$794$	2.24424	0.0796452
$795$	33.1982	1.17742
$796$	−11.0236	−0.390722
$797$	37.6312	1.33296	0.666482	−	0.745521i	$-0.267799\pi$
$797$	37.6312	1.33296	0.666482	+	0.745521i	$0.267799\pi$
$798$	35.5447	1.25827
$799$	4.68555	0.165763
$800$	−3.95431	−0.139806
$801$	−117.307	−4.14483
$802$	−25.6509	−0.905765
$803$	12.7981	0.451635
$804$	−27.9623	−0.986153
$805$	−5.48292	−0.193247
$806$	−6.26110	−0.220538
$807$	40.6033	1.42931
$808$	−16.9196	−0.595230
$809$	23.5674	0.828585	0.414292	−	0.910144i	$-0.364029\pi$
$809$	23.5674	0.828585	0.414292	+	0.910144i	$0.364029\pi$
$810$	37.0398	1.30144
$811$	−49.4319	−1.73579	−0.867894	−	0.496749i	$-0.834527\pi$
$811$	−49.4319	−1.73579	−0.867894	+	0.496749i	$0.834527\pi$
$812$	2.77563	0.0974054
$813$	−104.551	−3.66677
$814$	0	0
$815$	21.0689	0.738012
$816$	3.78778	0.132599
$817$	31.1977	1.09147
$818$	10.2179	0.357261
$819$	−6.98477	−0.244068
$820$	4.15492	0.145096
$821$	25.3706	0.885442	0.442721	−	0.896659i	$-0.354013\pi$
$821$	25.3706	0.885442	0.442721	+	0.896659i	$0.354013\pi$
$822$	−7.15545	−0.249575
$823$	−8.17024	−0.284797	−0.142398	−	0.989809i	$-0.545481\pi$
$823$	−8.17024	−0.284797	−0.142398	+	0.989809i	$0.545481\pi$
$824$	−19.1949	−0.668686
$825$	−12.0234	−0.418602
$826$	0.665958	0.0231716
$827$	0.990119	0.0344298	0.0172149	−	0.999852i	$-0.494520\pi$
$827$	0.990119	0.0344298	0.0172149	+	0.999852i	$0.494520\pi$
$828$	−27.1663	−0.944094
$829$	26.1236	0.907309	0.453654	−	0.891178i	$-0.350120\pi$
$829$	26.1236	0.907309	0.453654	+	0.891178i	$0.350120\pi$
$830$	−13.0139	−0.451721
$831$	−77.4607	−2.68708
$832$	−5.84247	−0.202551
$833$	3.18404	0.110320
$834$	14.1509	0.490006
$835$	−10.0289	−0.347066
$836$	−20.7651	−0.718174
$837$	−137.490	−4.75237
$838$	9.53901	0.329520
$839$	−40.1247	−1.38526	−0.692629	−	0.721294i	$-0.743548\pi$
$839$	−40.1247	−1.38526	−0.692629	+	0.721294i	$0.743548\pi$
$840$	−12.5739	−0.433842
$841$	−19.7056	−0.679505
$842$	−9.90361	−0.341301
$843$	71.5519	2.46438
$844$	13.4880	0.464276
$845$	12.5105	0.430376
$846$	74.0035	2.54429
$847$	2.39791	0.0823930
$848$	19.4761	0.668812
$849$	46.1443	1.58367
$850$	0.648708	0.0222505
$851$	0	0
$852$	1.26027	0.0431763
$853$	32.6763	1.11882	0.559408	−	0.828892i	$-0.311029\pi$
$853$	32.6763	1.11882	0.559408	+	0.828892i	$0.311029\pi$
$854$	−1.26161	−0.0431714
$855$	63.2552	2.16328
$856$	−1.13827	−0.0389054
$857$	9.95029	0.339896	0.169948	−	0.985453i	$-0.445640\pi$
$857$	9.95029	0.339896	0.169948	+	0.985453i	$0.445640\pi$
$858$	−9.42816	−0.321872
$859$	−4.53739	−0.154814	−0.0774068	−	0.997000i	$-0.524664\pi$
$859$	−4.53739	−0.154814	−0.0774068	+	0.997000i	$0.524664\pi$
$860$	−2.99149	−0.102009
$861$	22.8425	0.778470
$862$	1.69320	0.0576706
$863$	3.45484	0.117604	0.0588020	−	0.998270i	$-0.481272\pi$
$863$	3.45484	0.117604	0.0588020	+	0.998270i	$0.481272\pi$
$864$	−68.0913	−2.31651
$865$	13.3032	0.452324
$866$	−7.68698	−0.261214
$867$	−55.6611	−1.89035
$868$	7.26947	0.246742
$869$	−13.7303	−0.465768
$870$	−11.4130	−0.386936
$871$	−7.87543	−0.266849
$872$	−9.56659	−0.323966
$873$	83.6592	2.83144
$874$	38.9363	1.31704
$875$	−1.22417	−0.0413846
$876$	8.83114	0.298377
$877$	−23.2726	−0.785861	−0.392931	−	0.919568i	$-0.628539\pi$
$877$	−23.2726	−0.785861	−0.392931	+	0.919568i	$0.628539\pi$
$878$	21.4504	0.723917
$879$	−23.7386	−0.800684
$880$	−7.05369	−0.237780
$881$	12.3245	0.415224	0.207612	−	0.978211i	$-0.433431\pi$
$881$	12.3245	0.415224	0.207612	+	0.978211i	$0.433431\pi$
$882$	50.2887	1.69331
$883$	−17.6556	−0.594157	−0.297078	−	0.954853i	$-0.596012\pi$
$883$	−17.6556	−0.594157	−0.297078	+	0.954853i	$0.596012\pi$
$884$	−0.301140	−0.0101284
$885$	1.62109	0.0544922
$886$	2.52963	0.0849845
$887$	40.8268	1.37083	0.685415	−	0.728153i	$-0.259621\pi$
$887$	40.8268	1.37083	0.685415	+	0.728153i	$0.259621\pi$
$888$	0	0
$889$	21.7286	0.728755
$890$	−16.1218	−0.540404
$891$	−118.962	−3.98537
$892$	2.86017	0.0957657
$893$	62.7910	2.10122
$894$	2.06837	0.0691768
$895$	9.15637	0.306064
$896$	−1.77697	−0.0593644
$897$	−10.4657	−0.349441
$898$	−27.3237	−0.911804
$899$	24.3423	0.811861
$900$	−6.06543	−0.202181
$901$	5.75273	0.191651
$902$	22.5414	0.750546
$903$	−16.4463	−0.547299
$904$	−35.9571	−1.19592
$905$	−21.1056	−0.701573
$906$	−34.2844	−1.13902
$907$	57.3101	1.90295	0.951475	−	0.307725i	$-0.0995677\pi$
$907$	57.3101	1.90295	0.951475	+	0.307725i	$0.0995677\pi$
$908$	−10.2734	−0.340933
$909$	−44.8705	−1.48826
$910$	−0.959935	−0.0318215
$911$	21.6237	0.716426	0.358213	−	0.933640i	$-0.383386\pi$
$911$	21.6237	0.716426	0.358213	+	0.933640i	$0.383386\pi$
$912$	50.7600	1.68083
$913$	41.7972	1.38329
$914$	−15.7507	−0.520986
$915$	−3.07103	−0.101525
$916$	−0.448460	−0.0148175
$917$	1.40860	0.0465162
$918$	11.1704	0.368679
$919$	−30.6901	−1.01237	−0.506186	−	0.862424i	$-0.668945\pi$
$919$	−30.6901	−1.01237	−0.506186	+	0.862424i	$0.668945\pi$
$920$	−13.7737	−0.454107
$921$	−78.9524	−2.60157
$922$	−29.6099	−0.975149
$923$	0.354949	0.0116833
$924$	10.9466	0.360117
$925$	0	0
$926$	28.1806	0.926071
$927$	−50.9045	−1.67192
$928$	12.0554	0.395737
$929$	−8.88293	−0.291439	−0.145720	−	0.989326i	$-0.546550\pi$
$929$	−8.88293	−0.291439	−0.145720	+	0.989326i	$0.546550\pi$
$930$	−29.8910	−0.980165
$931$	42.6693	1.39843
$932$	−17.3601	−0.568650
$933$	64.6652	2.11704
$934$	−15.7263	−0.514582
$935$	−2.08347	−0.0681368
$936$	−17.5466	−0.573527
$937$	0.612588	0.0200124	0.0100062	−	0.999950i	$-0.496815\pi$
$937$	0.612588	0.0200124	0.0100062	+	0.999950i	$0.496815\pi$
$938$	−15.4456	−0.504317
$939$	−20.4089	−0.666019
$940$	−6.02092	−0.196381
$941$	33.3892	1.08846	0.544229	−	0.838937i	$-0.316822\pi$
$941$	33.3892	1.08846	0.544229	+	0.838937i	$0.316822\pi$
$942$	10.2636	0.334406
$943$	25.0221	0.814831
$944$	0.951030	0.0309534
$945$	−21.0797	−0.685722
$946$	−16.2295	−0.527667
$947$	10.3642	0.336791	0.168396	−	0.985720i	$-0.446141\pi$
$947$	10.3642	0.336791	0.168396	+	0.985720i	$0.446141\pi$
$948$	−9.47438	−0.307714
$949$	2.48724	0.0807394
$950$	8.69333	0.282049
$951$	−111.024	−3.60018
$952$	−2.17887	−0.0706176
$953$	14.8894	0.482316	0.241158	−	0.970486i	$-0.422473\pi$
$953$	14.8894	0.482316	0.241158	+	0.970486i	$0.422473\pi$
$954$	90.8585	2.94165
$955$	−7.51240	−0.243096
$956$	−20.8937	−0.675749
$957$	36.6554	1.18490
$958$	−7.48959	−0.241978
$959$	2.33986	0.0755582
$960$	−27.8924	−0.900224
$961$	32.7533	1.05656
$962$	0	0
$963$	−3.01868	−0.0972757
$964$	0.0270745	0.000872012 0
$965$	16.5435	0.532555
$966$	−20.5258	−0.660408
$967$	−43.3076	−1.39268	−0.696339	−	0.717713i	$-0.745189\pi$
$967$	−43.3076	−1.39268	−0.696339	+	0.717713i	$0.745189\pi$
$968$	6.02382	0.193613
$969$	14.9932	0.481650
$970$	11.4975	0.369163
$971$	8.91383	0.286058	0.143029	−	0.989718i	$-0.454316\pi$
$971$	8.91383	0.286058	0.143029	+	0.989718i	$0.454316\pi$
$972$	−43.6684	−1.40066
$973$	−4.62741	−0.148348
$974$	−8.14340	−0.260931
$975$	−2.33669	−0.0748340
$976$	−1.80166	−0.0576697
$977$	−13.3726	−0.427829	−0.213914	−	0.976852i	$-0.568621\pi$
$977$	−13.3726	−0.427829	−0.213914	+	0.976852i	$0.568621\pi$
$978$	78.8736	2.52210
$979$	51.7788	1.65486
$980$	−4.09149	−0.130698
$981$	−25.3704	−0.810015
$982$	−0.500593	−0.0159746
$983$	−23.1318	−0.737789	−0.368894	−	0.929471i	$-0.620264\pi$
$983$	−23.1318	−0.737789	−0.368894	+	0.929471i	$0.620264\pi$
$984$	57.3830	1.82930
$985$	−1.74676	−0.0556565
$986$	−1.97769	−0.0629826
$987$	−33.1012	−1.05362
$988$	−4.03558	−0.128389
$989$	−18.0156	−0.572863
$990$	−32.9063	−1.04583
$991$	−14.4144	−0.457887	−0.228944	−	0.973440i	$-0.573527\pi$
$991$	−14.4144	−0.457887	−0.228944	+	0.973440i	$0.573527\pi$
$992$	31.5735	1.00246
$993$	−4.63374	−0.147047
$994$	0.696142	0.0220803
$995$	−14.8223	−0.469899
$996$	28.8416	0.913880
$997$	1.01895	0.0322706	0.0161353	−	0.999870i	$-0.494864\pi$
$997$	1.01895	0.0322706	0.0161353	+	0.999870i	$0.494864\pi$
$998$	−11.8859	−0.376243
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6845.2.a.k.1.3		7
37.11	even	6		185.2.e.a.121.3	yes	14
37.27	even	6		185.2.e.a.26.3	✓	14
37.36	even	2		6845.2.a.l.1.5		7
185.27	odd	12		925.2.o.b.174.10		28
185.48	odd	12		925.2.o.b.824.10		28
185.64	even	6		925.2.e.c.26.5		14
185.122	odd	12		925.2.o.b.824.5		28
185.138	odd	12		925.2.o.b.174.5		28
185.159	even	6		925.2.e.c.676.5		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.e.a.26.3	✓	14	37.27	even	6
185.2.e.a.121.3	yes	14	37.11	even	6
925.2.e.c.26.5		14	185.64	even	6
925.2.e.c.676.5		14	185.159	even	6
925.2.o.b.174.5		28	185.138	odd	12
925.2.o.b.174.10		28	185.27	odd	12
925.2.o.b.824.5		28	185.122	odd	12
925.2.o.b.824.10		28	185.48	odd	12
6845.2.a.k.1.3		7	1.1	even	1	trivial
6845.2.a.l.1.5		7	37.36	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6845 = 5 \cdot 37^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6845.a (trivial)

\(f(q)\)	\(=\)	\(q-1.12084 q^{2} +3.33999 q^{3} -0.743718 q^{4} -1.00000 q^{5} -3.74360 q^{6} +1.22417 q^{7} +3.07527 q^{8} +8.15556 q^{9} +1.12084 q^{10} -3.59983 q^{11} -2.48401 q^{12} -0.699609 q^{13} -1.37210 q^{14} -3.33999 q^{15} -1.95945 q^{16} -0.578769 q^{17} -9.14107 q^{18} -7.75609 q^{19} +0.743718 q^{20} +4.08873 q^{21} +4.03484 q^{22} +4.47887 q^{23} +10.2714 q^{24} +1.00000 q^{25} +0.784150 q^{26} +17.2195 q^{27} -0.910439 q^{28} -3.04867 q^{29} +3.74360 q^{30} -7.98457 q^{31} -3.95431 q^{32} -12.0234 q^{33} +0.648708 q^{34} -1.22417 q^{35} -6.06543 q^{36} +8.69333 q^{38} -2.33669 q^{39} -3.07527 q^{40} +5.58669 q^{41} -4.58281 q^{42} -4.02235 q^{43} +2.67726 q^{44} -8.15556 q^{45} -5.02010 q^{46} -8.09571 q^{47} -6.54454 q^{48} -5.50140 q^{49} -1.12084 q^{50} -1.93309 q^{51} +0.520312 q^{52} -9.93959 q^{53} -19.3003 q^{54} +3.59983 q^{55} +3.76466 q^{56} -25.9053 q^{57} +3.41707 q^{58} -0.485356 q^{59} +2.48401 q^{60} +0.919473 q^{61} +8.94942 q^{62} +9.98382 q^{63} +8.35104 q^{64} +0.699609 q^{65} +13.4763 q^{66} +11.2569 q^{67} +0.430441 q^{68} +14.9594 q^{69} +1.37210 q^{70} -0.507354 q^{71} +25.0805 q^{72} -3.55519 q^{73} +3.33999 q^{75} +5.76834 q^{76} -4.40682 q^{77} +2.61906 q^{78} +3.81414 q^{79} +1.95945 q^{80} +33.0464 q^{81} -6.26179 q^{82} -11.6109 q^{83} -3.04086 q^{84} +0.578769 q^{85} +4.50841 q^{86} -10.1825 q^{87} -11.0705 q^{88} -14.3837 q^{89} +9.14107 q^{90} -0.856443 q^{91} -3.33102 q^{92} -26.6684 q^{93} +9.07400 q^{94} +7.75609 q^{95} -13.2074 q^{96} +10.2579 q^{97} +6.16619 q^{98} -29.3586 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{3} + 8 q^{4} - 7 q^{5} + 2 q^{6} - 3 q^{8} + 13 q^{9} - q^{11} - 6 q^{12} + 4 q^{13} - 10 q^{14} - 2 q^{15} - 2 q^{16} - 3 q^{17} - 6 q^{18} - 14 q^{19} - 8 q^{20} - q^{21} + 7 q^{22} + 15 q^{24}+ \cdots - 56 q^{99}+O(q^{100}) \) 7 * q + 2 * q^3 + 8 * q^4 - 7 * q^5 + 2 * q^6 - 3 * q^8 + 13 * q^9 - q^11 - 6 * q^12 + 4 * q^13 - 10 * q^14 - 2 * q^15 - 2 * q^16 - 3 * q^17 - 6 * q^18 - 14 * q^19 - 8 * q^20 - q^21 + 7 * q^22 + 15 * q^24 + 7 * q^25 - 11 * q^26 - 7 * q^27 - 9 * q^28 + 6 * q^29 - 2 * q^30 + 4 * q^32 + 11 * q^33 - 33 * q^34 + 20 * q^36 - 2 * q^38 - 25 * q^39 + 3 * q^40 + 7 * q^41 - 29 * q^42 - 19 * q^43 + 19 * q^44 - 13 * q^45 + 14 * q^46 + 4 * q^47 + 18 * q^48 - 5 * q^49 - 29 * q^51 - 8 * q^52 + 6 * q^53 - 9 * q^54 + q^55 - 25 * q^56 - 44 * q^57 + 12 * q^58 + 2 * q^59 + 6 * q^60 - 4 * q^61 + 25 * q^62 + 5 * q^63 - 31 * q^64 - 4 * q^65 + q^66 + 8 * q^67 - 13 * q^68 - 11 * q^69 + 10 * q^70 - 33 * q^71 + 43 * q^72 + 2 * q^75 - 12 * q^76 + 5 * q^77 + 22 * q^78 - 9 * q^79 + 2 * q^80 + 35 * q^81 - 76 * q^82 + 11 * q^83 - 40 * q^84 + 3 * q^85 - 17 * q^86 - 29 * q^87 - 20 * q^88 - 12 * q^89 + 6 * q^90 - 24 * q^91 - 7 * q^92 - 47 * q^93 + 10 * q^94 + 14 * q^95 - 15 * q^96 - 15 * q^97 + 54 * q^98 - 56 * q^99

Introduction

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